A CR Poincaré inequality on the complex sphere
type: Item
instance of: scholarly article
title: A CR Poincaré inequality on the complex sphere
zbMATH Open document ID: 1115.32022
author: Lijing Sun
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-14T00:00:00Z
review text: Let \(f\) be a smooth function on the unit sphere \( \mathbb{S}\) in \( \mathbb{C}^n\) and denote by \(\nabla_{\mathbb{C}}f\) its complex tangential gradient: this is the unique complex tangential vector field with the property that \(Xf(z) = \langle Xf(z),(\nabla_{\mathbb{C}}f)(z)\rangle \) for all complex tangential vector fields \(X\), where \( \langle\cdot, \cdot\rangle\) denotes the usual Hermitian Riemannian metric on complex-valued tangent vectors. It is proved that for \(1 \leq p <2n\) and \(q=2np/(2n-p),\) there exists a positive constant, depending only on \(n\) and \(p,\) such that \[ \| f-(f)_{\mathbb{S}}\| _{L^q(\mathbb{S})} \leq C \| \nabla_{\mathbb{C}}f\| _{L^p(\mathbb{S})} \] where, with \(d \sigma \) the normalized volume element on \(\mathbb{S}, (f)_{\mathbb{S}} \) is the integral mean \( \int_{\mathbb{S}} f \,d\sigma.\)
Mathematics Subject Classification ID: 32V20
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 5125956
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: complex tangential gradient
zbMATH Keywords: CR manifold
reviewed by: W. D. Evans
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.03.066
OpenAlex ID: W2058092994
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DOI: 10.1016/J.JMAA.2006.03.066
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description: scientific article; zbMATH DE number 5125956